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      【NOI2019模拟赛（五十一）】网格
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        <h3 id="题意简述"><a class="markdownIt-Anchor" href="#题意简述"></a> 题意简述</h3>
<p>你有一个<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n\times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.58333em;"></span><span class="strut bottom" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span><span class="mbin">×</span><span class="mord mathit">m</span></span></span></span>的网格，其中有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>个点是特殊的，你需要找两条<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(n,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mpunct">,</span><span class="mord mathit">m</span><span class="mclose">)</span></span></span></span>的路径，满足：</p>
<a id="more"></a>
<ul>
<li>除起点和终点外，两条路径不相交</li>
<li>两条路径经过的特殊点总数不超过<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.01968em;">l</span><span class="mord mathit">i</span><span class="mord mathit">m</span></span></span></span></li>
</ul>
<p>求方案数，对<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">p</span></span></span></span>取模，不保证模数为质数</p>
<h3 id="题解"><a class="markdownIt-Anchor" href="#题解"></a> 题解</h3>
<p>先来考虑一条路径，起点和终点视为特殊点，然后将所有特殊点排个序</p>
<p>设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_{i,j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10764em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>表示当前考虑到第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span>个特殊点，目前为止经过了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span>个特殊点的方案数；<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>g</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">g_{i,j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>表示从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span>号特殊点出发，不经过任何特殊点，到达<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span>号特殊点的方案数；<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">h_{i,j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>表示从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span>号特殊点出发到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span>号特殊点的方案数。不难得出如下转移：</p>
<ul>
<li>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>f</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>g</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_{i,j}=\sum_{k=1}^{i-1}f_{k,j-1}g_{k,i}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.8116690000000002em;"></span><span class="strut bottom" style="height:3.1137820000000005em;vertical-align:-1.302113em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10764em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.250005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">i</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10764em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mpunct">,</span><span class="mord mathit">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></p>
</li>
<li>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>g</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>h</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>−</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>g</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>k</mi></mrow></msub><msub><mi>h</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">g_{i,j}=h_{i,j}-\sum_{k=i+1}^{j-1}g_{i,k}h_{k,j}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.8587770000000001em;"></span><span class="strut bottom" style="height:3.219221em;vertical-align:-1.360444em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mrel">=</span><span class="mord mathit">i</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol large-op mop">∑</span></span></span><span style="top:-1.297113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></p>
</li>
<li>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mrow><msub><mi>x</mi><mi>j</mi></msub><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><msub><mi>y</mi><mi>j</mi></msub><mo>−</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>j</mi></msub><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">h_{i,j}=\binom{x_j-x_i+y_j-y_i}{x_j-x_i}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.45em;"></span><span class="strut bottom" style="height:2.4221079999999997em;vertical-align:-0.972108em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist"><span style="top:0.6859999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p>
</li>
</ul>
<p>这些转移可以<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>k</mi><mn>3</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">O(k^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span>完成</p>
<p>现在是两条路径。我们可以认为，第一条路径是从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1,2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathrm">2</span><span class="mclose">)</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mi>m</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(n-1,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathit">m</span><span class="mclose">)</span></span></span></span>，第二条路径是从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(n,m-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mpunct">,</span><span class="mord mathit">m</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span>，相当于是硬点了第一步和最后一步。然后两条路径分别做一遍上述<code>dp</code>，那么方案数显然就是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mo>∑</mo><mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>≤</mo><mi>l</mi><mi>i</mi><mi>m</mi><mo>+</mo><mn>4</mn></mrow></msub><msub><mi>f</mi><mrow><mn>1</mn><mo separator="true">,</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo separator="true">,</mo><mi>i</mi></mrow></msub><msub><mi>f</mi><mrow><mn>2</mn><mo separator="true">,</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo separator="true">,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\sum\limits_{i+j\leq lim+4}f_{1,k+2,i}f_{2,k+2,j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.7500050000000001em;"></span><span class="strut bottom" style="height:1.888226em;vertical-align:-1.138221em;"></span><span class="base textstyle uncramped"><span class="mop op-limits"><span class="vlist"><span style="top:0.9021129999999998em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mrel">≤</span><span class="mord mathit" style="margin-right:0.01968em;">l</span><span class="mord mathit">i</span><span class="mord mathit">m</span><span class="mbin">+</span><span class="mord mathrm">4</span></span></span></span><span style="top:-0.000005000000000088267em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="op-symbol small-op mop">∑</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10764em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">+</span><span class="mord mathrm">2</span><span class="mpunct">,</span><span class="mord mathit">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10764em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">+</span><span class="mord mathrm">2</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>。其中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi><mo>+</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">lim+4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.01968em;">l</span><span class="mord mathit">i</span><span class="mord mathit">m</span><span class="mbin">+</span><span class="mord mathrm">4</span></span></span></span>和<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k+2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">+</span><span class="mord mathrm">2</span></span></span></span>是因为起点和终点被算作了特殊点</p>
<p>这样得出的答案，不能保证路径不相交，于是我们要算出相交的方案，然后用总的减掉它。我们将两条路径第一次相交之后的部分交换，于是第一条路径终止于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(n,m-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mpunct">,</span><span class="mord mathit">m</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span>，第二条路径终止于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mi>m</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(n-1,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathit">m</span><span class="mclose">)</span></span></span></span>。不难发现，通过上述变换，我们可以将不合法方案与两条<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1,2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathrm">2</span><span class="mclose">)</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(n,m-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mpunct">,</span><span class="mord mathit">m</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mpunct">,</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mi>m</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(n-1,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathit">m</span><span class="mclose">)</span></span></span></span>的路径建立一一对应的关系，于是就可以和上面一样求出不合法方案数了</p>
<p>至于组合数的非质数模，可以先将模数分解，然后预处理不含模数因子的阶乘及其逆元，再记录模数因子的幂次即可。求组合数时，要枚举模数因子，然后乘上相应的幂次，单次求解复杂度<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>log</mi><mn>2</mn></msup><mi>p</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">O(\log^2 p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">p</span><span class="mclose">)</span></span></span></span></p>
<div class="highlight-box" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false" contenteditable="true" data-rel="CPP"><figure class="iseeu highlight /cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span 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class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br><span class="line">112</span><br><span class="line">113</span><br><span class="line">114</span><br><span class="line">115</span><br><span class="line">116</span><br><span class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br><span class="line">120</span><br><span class="line">121</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N=<span class="number">210</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Point</span>&#123;</span><span class="keyword">int</span> x,y;&#125; pt[N];</span><br><span class="line"><span class="keyword">int</span> g[N][N],h[N][N];</span><br><span class="line"><span class="keyword">int</span> T,n,m,tot,lim,p;</span><br><span class="line"><span class="keyword">int</span> s1[N][N],s2[N][N];</span><br><span class="line"></span><br><span class="line"><span class="keyword">namespace</span> Combo</span><br><span class="line">&#123;</span><br><span class="line">    <span class="keyword">const</span> <span class="keyword">int</span> N=<span class="number">400010</span>;</span><br><span class="line">    <span class="keyword">int</span> fac[N],ifac[N],fat[<span class="number">20</span>],fatp[N][<span class="number">20</span>],cnt;</span><br><span class="line">    </span><br><span class="line">    <span class="function"><span class="keyword">int</span> <span class="title">Pow</span><span class="params">(<span class="keyword">int</span> a,<span class="keyword">int</span> b)</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> ans=<span class="number">1</span>;</span><br><span class="line">        <span class="keyword">for</span>(;b;b&gt;&gt;=<span class="number">1</span>,a=<span class="number">1l</span>l*a*a%p)</span><br><span class="line">            <span class="keyword">if</span>(b&amp;<span class="number">1</span>) ans=<span class="number">1l</span>l*ans*a%p;</span><br><span class="line">        <span class="keyword">return</span> ans;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">int</span> <span class="title">exgcd</span><span class="params">(<span class="keyword">int</span> a,<span class="keyword">int</span> b,<span class="keyword">int</span> &amp;x,<span class="keyword">int</span> &amp;y)</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">if</span>(!b)&#123;x=<span class="number">1</span>;y=<span class="number">0</span>;<span class="keyword">return</span> a;&#125;</span><br><span class="line">        <span class="keyword">int</span> g=exgcd(b,a%b,x,y);</span><br><span class="line">        <span class="keyword">int</span> t=x;x=y;y=t-(a/b)*y;</span><br><span class="line">        <span class="keyword">return</span> g;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">int</span> <span class="title">inv</span><span class="params">(<span class="keyword">int</span> a)</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> x,y;</span><br><span class="line">        exgcd(a,p,x,y);</span><br><span class="line">        <span class="keyword">return</span> (x%p+p)%p;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">void</span> <span class="title">prework</span><span class="params">(<span class="keyword">int</span> n)</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> x=p;cnt=<span class="number">0</span>;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">2</span>;i*i&lt;=x;i++)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">if</span>(x%i) <span class="keyword">continue</span>;</span><br><span class="line">            fat[++cnt]=i;</span><br><span class="line">            <span class="keyword">while</span>(x%i==<span class="number">0</span>) x/=i;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span>(x&gt;<span class="number">1</span>) fat[++cnt]=x;</span><br><span class="line">        fac[<span class="number">0</span>]=ifac[<span class="number">0</span>]=<span class="number">1</span>;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n;i++)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">int</span> x=i;</span><br><span class="line">            <span class="built_in">memcpy</span>(fatp[i],fatp[i<span class="number">-1</span>],<span class="keyword">sizeof</span>(fatp[i]));</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">1</span>;j&lt;=cnt;j++)</span><br><span class="line">                <span class="keyword">while</span>(x%fat[j]==<span class="number">0</span>) x/=fat[j],fatp[i][j]++;</span><br><span class="line">            fac[i]=<span class="number">1l</span>l*fac[i<span class="number">-1</span>]*x%p;</span><br><span class="line">            ifac[i]=inv(fac[i]);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">int</span> <span class="title">C</span><span class="params">(<span class="keyword">int</span> n,<span class="keyword">int</span> m)</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">if</span>(n&lt;<span class="number">0</span>||m&lt;<span class="number">0</span>||n&lt;m) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">        <span class="keyword">int</span> res=<span class="number">1l</span>l*fac[n]*ifac[m]%p*ifac[n-m]%p;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=cnt;i++)</span><br><span class="line">            res=<span class="number">1l</span>l*res*Pow(fat[i],fatp[n][i]-fatp[m][i]-fatp[n-m][i])%p;</span><br><span class="line">        <span class="keyword">return</span> res;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">add</span><span class="params">(<span class="keyword">const</span> <span class="keyword">int</span> &amp;x,<span class="keyword">const</span> <span class="keyword">int</span> &amp;y)</span></span>&#123;<span class="keyword">return</span> (x+y&gt;=p)?(x+y-p):(x+y);&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">mns</span><span class="params">(<span class="keyword">const</span> <span class="keyword">int</span> &amp;x,<span class="keyword">const</span> <span class="keyword">int</span> &amp;y)</span></span>&#123;<span class="keyword">return</span> (x-y&lt;<span class="number">0</span>)?(x-y+p):(x-y);&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">dp</span><span class="params">(<span class="keyword">int</span> x1,<span class="keyword">int</span> y1,<span class="keyword">int</span> x2,<span class="keyword">int</span> y2,<span class="keyword">int</span> f[N][N])</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    pt[<span class="number">1</span>].x=x1;pt[<span class="number">1</span>].y=y1;</span><br><span class="line">    pt[tot].x=x2;pt[tot].y=y2;</span><br><span class="line">    <span class="built_in">memset</span>(h,<span class="number">0</span>,<span class="keyword">sizeof</span>(h));</span><br><span class="line">    <span class="built_in">memset</span>(g,<span class="number">0</span>,<span class="keyword">sizeof</span>(g));</span><br><span class="line">    <span class="built_in">memset</span>(f,<span class="number">0</span>,<span class="keyword">sizeof</span>(g));</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=tot;i++)</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">1</span>;j&lt;=tot;j++)</span><br><span class="line">            h[i][j]=Combo::C(pt[j].x-pt[i].x+pt[j].y-pt[i].y,pt[j].x-pt[i].x);</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=tot;i++)</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=i;j&lt;=tot;j++)</span><br><span class="line">        &#123;</span><br><span class="line">            g[i][j]=h[i][j];</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> k=i+<span class="number">1</span>;k&lt;j;k++)</span><br><span class="line">                g[i][j]=mns(g[i][j],<span class="number">1l</span>l*g[i][k]*h[k][j]%p);</span><br><span class="line">        &#125;</span><br><span class="line">    f[<span class="number">1</span>][<span class="number">1</span>]=<span class="number">1</span>;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">2</span>;i&lt;=tot;i++)</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">1</span>;j&lt;=i;j++)</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> k=j<span class="number">-1</span>;k&lt;i;k++)</span><br><span class="line">                f[i][j]=add(f[i][j],<span class="number">1l</span>l*f[k][j<span class="number">-1</span>]*g[k][i]%p);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="built_in">scanf</span>(<span class="string">"%d"</span>,&amp;T);</span><br><span class="line">    <span class="keyword">while</span>(T--)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">scanf</span>(<span class="string">"%d%d%d%d%d"</span>,&amp;n,&amp;m,&amp;tot,&amp;lim,&amp;p);</span><br><span class="line">        Combo::prework((n+m)&lt;&lt;<span class="number">1</span>);</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=tot;i++)</span><br><span class="line">            <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>,&amp;pt[i+<span class="number">1</span>].x,&amp;pt[i+<span class="number">1</span>].y);</span><br><span class="line">        sort(pt+<span class="number">2</span>,pt+<span class="number">2</span>+tot,[](Point a,Point b)&#123;<span class="keyword">return</span> a.x&lt;b.x||a.x==b.x&amp;&amp;a.y&lt;b.y;&#125;);</span><br><span class="line">        <span class="keyword">int</span> ans=<span class="number">0</span>;tot+=<span class="number">2</span>;</span><br><span class="line">        dp(<span class="number">1</span>,<span class="number">2</span>,n<span class="number">-1</span>,m,s1);</span><br><span class="line">        dp(<span class="number">2</span>,<span class="number">1</span>,n,m<span class="number">-1</span>,s2);</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">2</span>;i&lt;=lim+<span class="number">2</span>;i++)</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">2</span>;i+j&lt;=lim+<span class="number">4</span>;j++)</span><br><span class="line">                ans=add(ans,<span class="number">1l</span>l*s1[tot][i]*s2[tot][j]%p);</span><br><span class="line">        dp(<span class="number">1</span>,<span class="number">2</span>,n,m<span class="number">-1</span>,s1);</span><br><span class="line">        dp(<span class="number">2</span>,<span class="number">1</span>,n<span class="number">-1</span>,m,s2);</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">2</span>;i&lt;=lim+<span class="number">2</span>;i++)</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">2</span>;i+j&lt;=lim+<span class="number">4</span>;j++)</span><br><span class="line">                ans=mns(ans,<span class="number">1l</span>l*s1[tot][i]*s2[tot][j]%p);</span><br><span class="line">        <span class="built_in">printf</span>(<span class="string">"%d\n"</span>,ans);</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></div>
      
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